§ Open mapping theorem
- Given a surjective continuous linear map , image of open unit ball is open.
- Immediate corollary: image of open set is open (translate/scale open unit ball around by linearity) to cover any open set with nbhds.
§ Quick intuition
- Intuition 1: If the map we bijective, then thm is reasonably believeable given bounded/continuous inverse theorem, since would be continuous, and thus would map open sets to open sets, which would mean that does the same.
- In more detail: suppose exists and is continuous. Then implies . Since is continuous, the iverse image of an open set ( ) is open, and thus is open.
§ Why surjective
- Consider the embedding from to .
- The full space is open in the domain of , but is not open in , since any epsilon ball around any point in the diagonal would leak out of the diagonal.
- Thus, not every continuous linear map maps open sets to open sets.